How are the following questions typically answered...

[dev3245]

How are the following questions typically answered, that is what free
toolkits, APIs, etc are used to get the answers to the following
questions:

1) Given a line segment and a point x what point y on the line segment
lies closest to the given point x?

2) Given a line segment and a point on the line segment, what point on
the line segment lies exactly 1000 units from the starting point?

3) Given a line (say a representation of a road) what is the average
gradient (say from a raster DEM) for that segment?

I would like to automate questions like these or incorporate them into
an algorthym that can run off-line. It seems that GIS tools typically
are graphical and interactive, made for non-programmers.

Thanks,

ryan